Ihor,
Oh, my goodness! Don't give up on your students' ability to make sense of
fractions! And study after study has shown that memorizing without some clue as
to the concept does not last very long.

1. One problem we teachers have to overcome is the impulse to do too much
"giving the best explanation possible" at the expense of too little
encouragement to students that they themselves can make sense of fractions.

If we as teachers do not take the time and trouble to dig into fractions
concepts with our students, we send them along into algebra with half-baked
notions and a firm belief that they are "no good at math." AND If we want our
students to make sense of fractions, then we also have to dig deeper into our
own understanding.

2. There are some excellent resources to help teachers guide their students
through activities that make sense of fractions. One of the best ones I know is
"Understanding Fractions" by Christine Losq (available from Great Source
Education Group). This resource not only provides concept-building activities
that dig into the meaning of fractions, but also shows you how students
responded, what insights they gained, what insights the teachers gained from the
students' work (misconceptions as well as discoveries).

The original query involved comparing the fractions 6/7 and 7/6. Before getting
into any kind of computational strategies, I ask kids to simply picture what
these fractions represent. I need to know that my students can translate 6/7 and
7/6 into meaningful and accurate pictures and have the language to describe
those pictures. For example, I first have students "help me see" the meaning of
6 out of 7 equal parts (they usually draw a pie and simulate 7 equal parts).
Then say, "show me 1 sixth, 2 sixths, 3 sixths" etc. until kids realize that 7/6
is 6/6 plus one more sixth, which is greater than 1. After that the comparison
is easy. I usually follow with some similar examples (comparing a fraction less
than 1 with a fraction that represents a quantity that is greater than 1). I
want them to recognize that there are many things to look for in a fraction:
all the possible the relationships between the numerator and the denominator;
the relative size of the fractional part represented by each denominator.
NOTE: I find number line models less useful because you already have to know a
lot about fractions in order to successfully situate a fraction on a number
line. And many fractions we run into in our textbooks (like sevenths) are really
hard to show on a number line for comparison purposes.

3. Be sure to emphasize the importance of language in developing an
understanding of fractions.
For example, when you teach division of a fraction by a fraction, start with a
whole number example to set up the language.
8÷4 = ? means "how many groups of 4 can I take out of 8?" (There are 2 groups of
4 in 8, so the quotient is 2 and means 2 groups of 8.) Be sure to remind
students that the = means two different ways to show the same quantity and that
the quotient is not just a random number but a unit of measure, in this case
"groups of 4."

Apply the same language to division of a whole number by a fraction and the
result makes sense. For example,

8 ÷ 1/2 means "how many halves can I take out of 8 wholes?" The answer, 16, now
makes sense because there are 16 halves in 8 wholes. When you have student model
this kind of division problem with simple manipulatives like paper fraction
wheels or fraction bars that they can cut up and mark, they will focus on the
underlying meaning of the numerical expression and of the operation.

The application of language to division of a fraction by a fraction also works.

3/4 ÷ 1/2 means "how many one-half size pieces can I take out of a 3/4 size
piece?" (The quotient, 1 1/2 (half-size pieces), then makes sense. Again,
give kids the time to model the logic of the division to find out that you can
take out 1 one-half size piece and half of a one-half size piece.)
It does take close attention to what the quotient means because the idea of a
"whole unit" has a double meaning when you get into division of fractions by
fractions: You are dealing with less than one whole unit, since you only have
3/4 to start. The divisor, 1/2, functions both as a quantity related to the
original whole and as the unit that defines the quotient (1 and 1/2 half-size
pieces are in a 3/4 size piece of a whole).

4. Pie models and pictures really do help to reinforce the language connection
and the underlying concept, although I have a hard time sometimes convincing my
stressed middle school colleagues that this is so. So don't skimp on the time it
takes to let the concept sink in.

5. If we take the time to really explore fractions concepts, students will be
able to answer for themselves why, when you divide a whole number by a fraction
you seem to end up with "more than what you started with" and why, when you
divide a fraction by a fraction, the quotient sometimes appears to be "more"
than what you started with.

7. The mental math involved is the ultimate goal, of course. And the "invert
and multiply" rule is the shortcut we want all of our students to eventually be
masterful with. However, if we just "give the best explanation possible" and
focus too quickly on the underlying factoring and computational skills, we will
NEVER help our student truly understand fractions.

Most important: Kids really will rise to the challenge if you constantly
communicate your belief in their abilities to dig into and understand the
language and logic of math.

This material is based upon work supported by the
National Science Foundation under Grant DUE-0226284.
Any opinions, findings, and conclusions or recommendations
expressed in this material are those of the author(s)
and do not necessarily reflect the views of the
National Science Foundation.